Capturing a phylogenetic tree when the number of character states varies with the number of leaves

We show that for any two values $\alpha, \beta >0$ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha \rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^\beta \rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.Comment: 3 pages, 0 figure

Tracing evolutionary links between species

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM

An Arrow-type result for inferring a species tree from gene trees

The reconstruction of a central tendency `species tree' from a large number of conflicting gene trees is a central problem in systematic biology. Moreover, it becomes particularly problematic when taxon coverage is patchy, so that not all taxa are present in every gene tree. Here, we list four desirable properties that a method for estimating a species tree from gene trees should have. We show that while these can be achieved when taxon coverage is complete (by the Adams consensus method), they cannot all be satisfied in the more general setting of partial taxon coverage.Comment: 5 pages, 0 figure

Expected Anomalies in the Fossil Record

The problem of intermediates in the fossil record has been frequently discussed ever since Darwin. The extent of `gaps' (missing transitional stages) has been used to argue against gradual evolution from a common ancestor. Traditionally, gaps have often been explained by the improbability of fossilization and the discontinuous selection of found fossils. Here we take an analytical approach and demonstrate why, under certain sampling conditions, we may not expect intermediates to be found. Using a simple null model, we show mathematically that the question of whether a taxon sampled from some time in the past is likely to be morphologically intermediate to other samples (dated earlier and later) depends on the shape and dimensions of the underlying phylogenetic tree that connects the taxa, and the times from which the fossils are sampled